Lecture4_Spring11

Lecture4_Spring11 - Elec 210 Lecture 4 Conditional...

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Elec210 Lecture 4 1 Elec 210: Lecture 4 Conditional Probability Properties Total Probability Theorem Bayes’ Rule Independence Example: Medical Testing Thus far, we have looked at the probability of events occurring “individually”, without regards to any other event. However, if we KNOW that a particular event occurs, then how does this change the probabilities of other events?
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Pizza Time Journal of Statistics Education v.6, n.1 (1998) You have pulled out a slice of pizza that contains pepperoni. What is the probability that it has mushrooms?
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Elec210 Lecture 4 3 Conditional Probability Knowledge that an event B has occurred may alter the probability that an event A occurs. The conditional probability , P[A|B], of event A given an event B, is defined as: P[A] is called the a priori probability P[A|B] is called the a posteriori probability [] B P B A P B A P = | A B S A B S where we assume that P [ B ] >0 Q: Assuming outcomes are selected at random, which is bigger in the two examples below: P[A] or P[A|B] ?
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Sample Space Interpretation
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Elec210 Lecture 4 5 Relative Frequency Interpretation If we interpret probability as relative frequency, then P[A|B] is the relative frequency of the event A B in experiments where B occurred . Suppose the experiment is performed n times, and suppose that the event B occurs n B times, and the event A B occurs n A B times, the relative frequency of interest is: [] B P B A P n n n n n n B B A B B A = A B S B A If B is known to have occurred, then A can occur only if occurs B A
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Example Prior information: An integer between 1 and 10 is chosen at random. Suppose we are interested in two events: A 2 = “The integer is 2” A 8 = “The integer is 8” Since the integer is chosen at random, our prior probabilities are Suppose we learn that the integer chosen is greater than 5, how does that change our probabilities? Let B = “The number is greater than 5” = {6,7,8,9,10} Elec210 Lecture 4 6 28 [ ] 0.1 [ ] 0.1 PA == 2 2 [] [{2} {6, 7,8, 9,10}] [{}] 0 [| ] 0 [ ] [{6,7,8,9,10}] [{6,7,8,9,10}] 0.5 B PP PA B PB P P = = = 8 8 [{8} {6,7,8,9,10}] [{8}] 0.1 ] 0 . 2 [ ] [{6,7,8,9,10}] [{6,7,8,9,10}] 0.5 B P P = = =
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Elec210 Lecture 4 7 Example An urn contains two black balls and three white balls. Two balls are selected at random without replacement. Find the conditional probability that the second ball is black given that the first ball is black. Solution Label the 5 balls as shown first letter = color second letter = identifier B 1 = “first ball black” B 2 = “second ball black” This is consistent with our intuition that if the first ball is black, then on the second draw there are four balls left, one of which is black. [] 4 1 8 2 | 1 2 1 1 2 = = = B P B B P B B P Ba Bb Wc Wd We Ba Ba,Bb Ba,Wc Ba,Wd Ba,We Bb Bb,Ba Bb,Wc Bb,Wd Bb,We Wc Wc,Ba Wc,Bb Wc,Wd Wc,We Wd Wd,Ba Wd,Bb Wd,Wc Wd,We We,Ba We,Bb We,Wc We,Wd FIRST BALL SECOND BALL B 1 B 2
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Elec210 Lecture 4 8 Elec 210: Lecture 4 Conditional Probability Properties Total Probability Theorem Bayes’ Rule Independence Example: Medical Testing
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Elec210 Lecture 4 9
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Lecture4_Spring11 - Elec 210 Lecture 4 Conditional...

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